Conceptual derivation of (classical mechanical) energy?

[Gerenuk]

In the classical sense energy could be seen as an auxiliary invariant, but I haven't checked if energy is more fundamental in special relativity. Maybe someone can comment on this?
That I want to check soon.

And I've explained it a couple of times: a mathematician knows that if he sees a gradient, he wants to dot it and integrate. That's a very good reason to do that.

 

[RoyLB]

In the book "Mathematical Aspects of Classical and Celestial Mechanics", 3rd edition, by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt, this paragraph appears on page 19(!)
------------
Example 1.1. A natural mechanical system is a triplet (M, T, V ), where M is a smooth configuration manifold, T is a Riemannian metric on M (the kinetic energy of the system), V is a smooth function on M (the potential of the force field). A Riemannian metric is a smooth function on the tangent bundle that is a positive definite quadratic form on each tangent plane. The Lagrange function is L = T + V (the function V : M → R is lifted to a function from TM into R in the obvious way).
-----------

Now, to me, the idea of kinetic energy being a "metric" makes sense since it is (typically? always?) a quadratic function of the coordinates, just like the (square) of the distance between two points in Euclidean space. So, is that all energy is - a length in some space? The rest of the terms used here don't quite make sense to me yet, and I don't think they're defined in this book (at least not in the pages prior to page 19). Do you think this line of attack makes any sense? If so, do you know of any good references that will explain some of these terms (manifold, tangent bundle, etc.?)

- Roy

 

[RoyLB]

And I've explained it a couple of times: a mathematician knows that if he sees a gradient, he wants to dot it and integrate. That's a very good reason to do that.

Gerenuk,

I don't dispute that the dot product makes sense from a mathematical standpoint. I would have thought it was obvious from what I had posted several times now, but my intended audience is not mathematicians, physicists nor engineers.

- Roy

 

[Cleonis]

[...]
Now, to me, the idea of kinetic energy being a "metric" makes sense since it is (typically? always?) a quadratic function of the coordinates, just like the (square) of the distance between two points in Euclidean space.
[...]

Well, angular momentum m r^2 \omega is a quadratic function of 'r', but there does not seem to be a way of interpreting that as akin to a metric.
I agree that interpreting kinetic energy as related to a metric is tempting, but I think it's a dead end.The challenge is, I suppose, to capture in words the most general sense of the concept of Energy.

It appears that materials have a state of zero energy, and that energy is accumulated when the material is pushed away from this ground state.

When an elastic material is deformed it stores energy, which is released when the material relaxes again. In elastic deformation the molecules of the material do not slide along each other (when they do slide you have plastic deformation, which dissipates energy). In the case of elastic deformation you are deforming the very molecules away from their ground state.

In a molecular bond there is a distance between the atoms that is the state of least energy. I suppose the most fundamental level of description we currently have is the following: given the quantum physics of molecular orbitals there is a state (of the molecule) that is the most probable. When a molecule is deformed away from that most probable state then it tends to return to the most probable state.

Another aspect: is it possible to view gravitational potential energy as a deformation away from a state of lowest energy? I think so. According to GR gravitational interaction is mediated by deformation of spacetime. (In other words: spacetime curvature acts as mediator of gravitational interaction.) I infer that when two objects are pulled apart then there is a net increase in spacetime deformation.

Finally, kinetic energy.
When two objects have a velocity relative to each other then that two-object system has the potential to do work.
Example, a electric car that is designed for regenerative braking. When the car and the Earth have a velocity relative to each other then regenerative braking will recharge the batteries. The state of lowest energy is where the two objects have no relative velocity.

So in all I like to think of the relative velocity of the two objects in a two-object system as a form of potential energy.
I don't so much think of the quadratic form, I think of the potential to do work, expressed as force acting over a certain distance.
What is special of course is that in the case of kinetic energy this potential is realized only at the very instant in time that the two objects actually interact. When two objects interact with each other, and not with other surrounding objects (as in a collision) then they are in effect a two-object system.

 

[Cleonis]

[...]
I realized that I was unable to bridge the gap in a satisfactory way between F=ma and W=Fd or T=1/2mv^2.
[...]

Hi Roy,

I believe I did the very thing you requested: I bridged from F=ma to W=Fd, using just d=1/2*a*t² and v=a*t. That was in https://www.physicsforums.com/showpost.php?p=2711586&postcount=24". I'm very curious what you think about that demonstration.

Cleonis

 

[Riad]

So you see in my view energy is just an artificial number that happens to be conserved in addition to momentum conservation, since we only have inverse square law forces.

Without inverse square laws, maybe our world wouldn't conserve energy.

(I haven't generalized my proof to non-instantaneous forces yet :( )

I like your derivation as it includes cross forces too. In my thinking about this subject I cam to the conclusion that one can postulate a modified Newtons law of action and reaction by saying that for equal point masses ' the displacement of one point mass must be offset by an opposite displacement of another' . as if the world is sitting on a knife edge and moving one particle must be offset by another. If we accept this (and I could not find a counter example yet) then differentiate once and you get conservation of momentum and differentiate a second time and you get acceleration or the sum of forces zero.. as Newton put it originally. Non equal masses can also be included.

 

[Acut]

@Riad
There's a counterexample. Just imagine 3 charged masses aligned (named 1, 2 and 3. 2 is the central one). Suppose charge of 1 = charge of 3, and distance 1-2 = distance 2-3. Than 1 will exert in 2 a force, but only 1 will be displaced.

You don't need to impose that those point masses have the same mass nor that they are equal. Just use Newton's definition F=dp/dt, and use Newton's third law in order to find that, regardless of the properties of thoses masses, or the quantity of bodies under study, we will always have conservation of momentum.

In fact, you should start backwars: conservation of momentum always holds (even in relativity or quantum mechanics), but there are situations in which Newton's third law doesn't apply. So you can't really use Newton's third law to find cons. of momentum, but use cons. of momentum from Newton's third law.

 

[Cleonis]

[...]
Is energy just a mathematical fiction / convenience, like imaginary numbers (OK - perhaps I opened a can of worms with that statement :-) or is there some physical "reality" to it?
[...]

In the case of nuclear physics the nuclear binding energy makes a difference. The mass of a nucleus is not just the sum of the constituent protons and neutrons, the internal energy has inertial mass too.

In the case of a very large nucleus, such as a Uranium nucleus, there is so much repulsive force between the protons that the nucleus is teetering on the brink of falling apart. A neutron absorption event can trigger nuclear fission. The combined mass of the two fission products is smaller than the mass of the original Uranium nucleus.

In a sense the Uranium nucleus is under very large internal stress. There is the strong nuclear force that acts to keep the nucleus together, and the Coulomb repulsion between the protons acts to make the nucleus fission. This internal stress is a form of potential energy, and it has a corresponding inertial mass. The magnitude of that inertial mass is given by the following expression:

m = \frac{E}{c^2}

 

[Studiot]

Roy,

I find the most successful learning method is recursive. I call it a spiral. Only (a coherent)part of a subject is presented on each circuit of the centre. As we move outwards the diameter expands and our knowledge expands and more is presented.

This is why the sort of treatise prepared by a professor is often poor (=heavy going) teaching material but an excellent reference once the subject has been mastered.

I commend you to the little monograph written by M B Glauert, entitled Principles of Dynamics.

This monograph is pitched at about the level you seem to require, and contains many explanatory insights, in particular his explanation of what is meant by a particle when introducing Newton's laws.

 

[RoyLB]

Hi Roy,

I believe I did the very thing you requested: I bridged from F=ma to W=Fd, using just d=1/2*a*t² and v=a*t. That was in https://www.physicsforums.com/showpost.php?p=2711586&postcount=24". I'm very curious what you think about that demonstration.

Cleonis

Cleonis,

My initial objection was that this seemed like pure mathematics to me. Acceleration, distance, and velocity are kinematically related (or if you prefer, are derivatives of one other), so of course acceleration times distance will yield velocity squared terms. Now that I've embraced that energy is all just mathematics at this level, your derivation is not too bad. I don't like that you multiplied both sides by a in equation (3) (which is OK for a mathematical proof), but this step is unnecessary. Just solve (2) for t and substitute into (1) and you can get to (4) pretty quickly.

Thanks
Roy

 

[RoyLB]

Cleonis

Well, angular momentum m r^2 \omega is a quadratic function of 'r', but there does not seem to be a way of interpreting that as akin to a metric.
I agree that interpreting kinetic energy as related to a metric is tempting, but I think it's a dead end.

Perhaps it only works for quadratic functions of the velocity coordinates, and not the position ones

The challenge is, I suppose, to capture in words the most general sense of the concept of Energy.

It appears that materials have a state of zero energy, and that energy is accumulated when the material is pushed away from this ground state.

When an elastic material is deformed it stores energy, which is released when the material relaxes again. In elastic deformation the molecules of the material do not slide along each other (when they do slide you have plastic deformation, which dissipates energy). In the case of elastic deformation you are deforming the very molecules away from their ground state.

In a molecular bond there is a distance between the atoms that is the state of least energy. I suppose the most fundamental level of description we currently have is the following: given the quantum physics of molecular orbitals there is a state (of the molecule) that is the most probable. When a molecule is deformed away from that most probable state then it tends to return to the most probable state.

Another aspect: is it possible to view gravitational potential energy as a deformation away from a state of lowest energy? I think so. According to GR gravitational interaction is mediated by deformation of spacetime. (In other words: spacetime curvature acts as mediator of gravitational interaction.) I infer that when two objects are pulled apart then there is a net increase in spacetime deformation.

Finally, kinetic energy.
When two objects have a velocity relative to each other then that two-object system has the potential to do work.
Example, a electric car that is designed for regenerative braking. When the car and the Earth have a velocity relative to each other then regenerative braking will recharge the batteries. The state of lowest energy is where the two objects have no relative velocity.

So in all I like to think of the relative velocity of the two objects in a two-object system as a form of potential energy.
I don't so much think of the quadratic form, I think of the potential to do work, expressed as force acting over a certain distance.
What is special of course is that in the case of kinetic energy this potential is realized only at the very instant in time that the two objects actually interact. When two objects interact with each other, and not with other surrounding objects (as in a collision) then they are in effect a two-object system.

I like this. As you indicate, potential energy is a function of the relative configuration in space of the components of the system. Kinetic energy is a function of the relative velocities of the components of the system.

If we do have conservative forces - that is if there exists a gradient of a potential function in space - then particles can be considered to be simply following a geodesic in this space. For every meter traveled on a non-flat gradient surface, the particle will accelerate a certain amount. This looks like a "force", but its "really" just a particle following a path in space. The distance traveled times that acceleration will yield velocity squared terms, and we have conservation of energy. Of course, I'm heavily borrowing from general relativity here. In this view, energy is the more fundamental concept, and forces are just derived from it.

- Roy

 

[RoyLB]

In the case of nuclear physics the nuclear binding energy makes a difference. The mass of a nucleus is not just the sum of the constituent protons and neutrons, the internal energy has inertial mass too.

In the case of a very large nucleus, such as a Uranium nucleus, there is so much repulsive force between the protons that the nucleus is teetering on the brink of falling apart. A neutron absorption event can trigger nuclear fission. The combined mass of the two fission products is smaller than the mass of the original Uranium nucleus.

In a sense the Uranium nucleus is under very large internal stress. There is the strong nuclear force that acts to keep the nucleus together, and the Coulomb repulsion between the protons acts to make the nucleus fission. This internal stress is a form of potential energy, and it has a corresponding inertial mass. The magnitude of that inertial mass is given by the following expression:

m = \frac{E}{c^2}

Cleonis

I think what you're saying is that to really appreciate energy, we have to go beyond the simplification of our Newtonian worldview and embrace relativity. Forget about this ineffable "potential energy stored in a spring". The compressed spring has more mass (imperceptible as it is to us) than the uncompressed one. In "realty", energy, mass, momentum are all aspects of the same thing.

- Roy

 

[RoyLB]

Roy,

I find the most successful learning method is recursive. I call it a spiral. Only (a coherent)part of a subject is presented on each circuit of the centre. As we move outwards the diameter expands and our knowledge expands and more is presented.

This is why the sort of treatise prepared by a professor is often poor (=heavy going) teaching material but an excellent reference once the subject has been mastered.

I commend you to the little monograph written by M B Glauert, entitled Principles of Dynamics.

This monograph is pitched at about the level you seem to require, and contains many explanatory insights, in particular his explanation of what is meant by a particle when introducing Newton's laws.

Studiot,

Thanks for the advice and the recommendation. I will look for it.

- Roy

 

[Cleonis]

[...] this step is unnecessary. Just solve (2) for t and substitute into (1) and you can get to (4) pretty quickly. [...]

Paraphrasing Goethe: apologies, my friend, for the unnecessarily long derivation, I didn't have time to write a shorter one.

About the relation between F=ma and E=1/2mv² :

I like the example of an electric car designed for regenerative braking. For simplicity let's assume that the electric generator of the car is 100% efficient in converting kinetic energy to electric energy. (With the electric energy converted to chemical potential energy in the car battery.)

The car is braking over a distance 'd' along the road, maintaining uniform deceleration. Then the level of energy in the battery increases linear with distance. Hence the level of kinetic energy of the car is decreasing linear with distance.

What intrigues me is that the quadratic expression E=1/2mv² expresses a linear relation when viewed as a function of distance travelled.

 

[Cleonis]

[...] In this view, energy is the more fundamental concept, and forces are just derived from it. [...]

I do think of the concept of 'force' as physics shorthand. When thinking in terms of Newtonian dynamics the concept of force is among the fundamental tools of the thinking process.

I find it tempting to speculate that 'probability' is the underlying theme of potential energy on one hand, and entropy on the other hand.

We know that increase of entropy can drive an endothermic process. A mix of ice and salt will decrease in temperature as the ice dissolves in the brine.

Right now I can't think of a counterexample, and I venture to say: any form of storing potential energy in elastic deformation or in chemical form is in one way or another a process of deforming away from a ground state. The ground state is the quantummechanically most probable state. Needless to say, this is highly speculative

 

[Riad]

@Riad
There's a counterexample. Just imagine 3 charged masses aligned (named 1, 2 and 3. 2 is the central one). Suppose charge of 1 = charge of 3, and distance 1-2 = distance 2-3. Than 1 will exert in 2 a force, but only 1 will be displaced.

You don't need to impose that those point masses have the same mass nor that they are equal. Just use Newton's definition F=dp/dt, and use Newton's third law in order to find that, regardless of the properties of thoses masses, or the quantity of bodies under study, we will always have conservation of momentum.

In fact, you should start backwars: conservation of momentum always holds (even in relativity or quantum mechanics), but there are situations in which Newton's third law doesn't apply. So you can't really use Newton's third law to find cons. of momentum, but use cons. of momentum from Newton's third law.

I can not quite get you.. if you move 1 then 3 moves also- otherwise you have the center of mass (of the three) moving without an ext. force involved. It seems to me that if it is possible to negate this ballanced displacement idea, then momentum is not conserved.. note that all events are taking the same time period dt- so velocity would be the same as momentum. The most interesting thing about this idea is that it can be used to justify the second and third laws of Newton and it can even explain inertia.. since to move one mass forward you need to push another back.. so that the sum of sum(distance x mass)= zero.

 

[Cleonis]

Cleonis
I think what you're saying is that to really appreciate energy, we have to go beyond the simplification of our Newtonian worldview and embrace relativity.

Actually no, that's kind of a step away from what I intended.

The question you raised was whether the concept of Energy can be seen as essentially a bookkeeping device, useful as a tool, but that we shouldn't attribute physical existence to it.

Well, the lesson from relativistic physics is that to Matter and Energy we have to attribute the same level of existence; they're on equal par, given that both matter and energy have inertial mass.

The nuclear binding energy example is, I guess, the only example where we can actually measure the mass difference that arises from difference in internal energy.
Theoretically a compressed spring has additional inertial mass, corresponding to the stored energy, but that mass difference is far, far too small to be measurable.

 

[GRDixon]

I have a pedagogical question, and a philosophical question, both involving energy:

What exactly is energy, anyway?
Direct answers or pointers to relevant references would be appreciated.

Thanks!
Roy

You might enjoy "The Feynman Lectures on Physics," V1, Sect. 4-1: "What is energy". Herewith a couple of quotes: "...there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes ... the energy has a large number of different forms, and there is a formula for each one ... we have no knowledge of what energy is ..."

 

[RoyLB]

All,

What do you guys think of this? I just found it in, from all places, Wikipedia:

(http://en.wikipedia.org/wiki/Energy#Regarding_applications_of_the_concept_of_energy)
In classical physics energy is considered a scalar quantity, the canonical conjugate to time. In special relativity energy is also a scalar (although not a Lorentz scalar but a time component of the energy-momentum 4-vector).[14] In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of space-time (= boosts).

So, of course I checked (http://en.wikipedia.org/wiki/Canonical_conjugate)
In physics, conjugate variables are pair of variables mathematically defined in such a way that they become Fourier transform duals of one-another, or more generally are related through Pontryagin duality.

Time and energy - as energy and frequency in quantum mechanics are directly proportional to each other.

So - energy is connected to time via Fourier transforms! As a controls guy I understand FTs. I also know they are important to the study of quantum mechanics, although I'm not that far along yet in my study of QM.

(I did check Pontryagin duality, but that entry quickly lost me)

- Roy

 

[RoyLB]

Actually no, that's kind of a step away from what I intended.

The question you raised was whether the concept of Energy can be seen as essentially a bookkeeping device, useful as a tool, but that we shouldn't attribute physical existence to it.

Well, the lesson from relativistic physics is that to Matter and Energy we have to attribute the same level of existence; they're on equal par, given that both matter and energy have inertial mass.

The nuclear binding energy example is, I guess, the only example where we can actually measure the mass difference that arises from difference in internal energy.
Theoretically a compressed spring has additional inertial mass, corresponding to the stored energy, but that mass difference is far, far too small to be measurable.

Cleonis,

We may agree.

I was lamenting the fact that energy appeared to be just a bookkeeping device in Newtonian physics, when I knew it was more than that. I think to really appreciate what energy is, you have to go beyond the approximation that is Newtonian physics and turn to relativity. From that vantage point, one can see that matter and energy are on equal par, as you say. Under Newton, matter and energy are two very different things, and one must view energy as essentially a fallout of the math under that approximation to "reality".

Or am I still misunderstanding your point?

- Roy

 

[Cleonis]

I think to really appreciate what energy is, you have to go beyond the approximation that is Newtonian physics and turn to relativity. From that vantage point, one can see that matter and energy are on equal par, as you say. Under Newton, matter and energy are two very different things, and one must view energy as essentially a fallout of the math under that approximation to "reality".

I guess I never gave that much thought. I am accustomed to the relativistic view on energy, and I guess I just brought that to bear on the Newtonian framework.

It may be that strictly within the theoretical framework of Newtonian theory it's possible to view energy as only a bookkeeping device, not atttributing physical reality to it - I don't know. Still, there is the empirical finding that we see conservation of energy. I think that that in itself is suggestive that energy is part of the physical world, independent of any theory we formulate.

 

[RoyLB]

You might enjoy "The Feynman Lectures on Physics," V1, Sect. 4-1: "What is energy". Herewith a couple of quotes: "...there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes ... the energy has a large number of different forms, and there is a formula for each one ... we have no knowledge of what energy is ..."

GRDixon,

I do indeed enjoy the Lectures and Feynman's other books. I was very disappointed that the great Feynman gave up when it came to explaining energy He explained quantum electrodynamics to the layman, but he couldn't explain energy to Caltech freshmen! IIRC, he promised in the lectures to derive the formula T=1/2mv^2, but I don't think he ever did

- Roy

 

[Studiot]

Isn't all this esoteric discussion is rather OTT for control engineering?
Does relativity have any relevance? (smile)

Here is an excerpt from my reference to whet you appetite for practical applied science.

…..The meaning of the word particle in the laws must be studied first. A particle is often said to be a point mass with no spatial extent. Atomic nuclei and electrons might be thought of as particles of this type, but Newton’s laws are not intended to apply to such small scale phenomena; usually quantum mechanics must be used instead. In classical mechanics the smallest piece of matter we need to consider contains enormous numbers of atoms and on this scale we can ignore atomic structure and think of matter as continuous.
Accordingly, we define a particle to be a material body whose dimensions, though not zero, are sufficiently small for the internal structure of the particle to be unimportant. The actual size permissible depends upon the particular physical problem. Thus the Earth may be treated as a single particle for the discussion of its movement around the sun, but a grain of sand cannot be treated as one in the formation of a sand dune. For our purposes the essential feature of a particle is that its position is sufficiently described by asingle vector r, the position vector from some origin

Glauert develops a similar argument to that presented here by Cleonis from this one fact right up to a complete derivation of total mechanical energy.

 

[RoyLB]

Studiot

Isn't all this esoteric discussion is rather OTT for control engineering?
Does relativity have any relevance? (smile)

For control engineering, I'd agree. For my own knowledge, its all relevant

Here is an excerpt from my reference to whet you appetite for practical applied science.



Glauert develops a similar argument to that presented here by Cleonis from this one fact right up to a complete derivation of total mechanical energy.

So what does he say about what energy is?

- Roy

 

[Studiot]

There's quite a few pages of it, as he develops a complete theory of kinetic, potential and rotational energy and the consequential total mechanical energy.

You have to take something as given and he starts from Newtons second law expressed as

F = mr\limits^{..}

 

[RoyLB]

There's quite a few pages of it, as he develops a complete theory of kinetic, potential and rotational energy and the consequential total mechanical energy.

You have to take something as given and he starts from Newtons second law expressed as

F = mr\limits^{..}

Studiot,

Nevermind the math (unless its truly novel). Does he opine at all as to the nature of energy?

- Roy

 

[Benson]

Hi All,

I'm currently teaching an introductory thermo and fluids course to 2nd year college diploma students who come through vocational high schools (so their mechanics conceptual fundamentals and mathematical fundamentals aren't great). I've started by giving them an overview of all the different forms of energy and some calculations for converting between them, and had similar problems explaining just why *is* energy equal to force x distance (I drew upon the a = F/m concept too).

I've now reached the point where I'm about to teach them calculating temperature rise in a closed system (no mass flow in or out) - e.g. a tank of liquid with a heating element of X Watts, a stirrer with a shaft power of Y Watts and heat loss of Z Watts, how long does it take to raise to a certain temperature. And of course with this you have the concept that part of the temperature rise is due to the heating element, and part is due to the stirrer doing work on the fluid. And then I got to thinking - how do I explain to them *how* or *why* - as in, through what mechanism - a spinning stirrer causes the temperature of a fluid to rise?

They understand the concept of friction, and intuitively that it dissipates heat - so would it be right to say that the temperature change (increase in internal energy) is due to increased friction between the water molecules as they move past/over/rub against each other with greater velocity?

Thanks,
Benson

 

[Philip Wood]

The following is naive, but, I would argue, logical. Can be made more sophisticated as required.

Starting point: Define the body's KE as the work it can do because of its motion.

Think of something like a boulder moving on a frictionless surface. Lasso it and exert a force F on it, so retarding it. Then

KE = Work done on rope by boulder as it slows down = ∫F.dx

But F = - \frac{d\textbf{v}}{dt}

This leads swiftly to the familiar formula.

 

[Riad]

In an isolated system all the input Heating Energy will go into increasing the internal temperature after some time. This increase involves the heat capacity of the liquid as is well known.
The input KE also becomes eventually internal energy and causes an additional increase in temperature to be calculated exactly the same way above as your units are all in watts in the two cases.
How the KE is converted to internal energy (expressed as an increase in temperature) is not difficult to explain. This conversion happens only if the fluid has a viscosity. Viscosity works like friction. Molecules of high velocity(KE) rubbing against others of smaller velocity causing them to accelerate(it is still a KE but not visible as it is only within the substance ie internal). This rubbing action happens between the fluid and blade material and also between the fluid and fluid of smaller velocity in the small eddies.

 

[Riad]

[Benson:Mar5-12, 07:58 AM Re: Conceptual derivation of (classical mechanical) energy? #57]
Just to complete my answer above;
To find the dynamic response of the container with stirrer, heating and losses;
Increase in temp per second=(heating power+stirrer power -lost power)/sum(mass.specific heat of all masses involved: rotor,liquid and container).
If you take this to be a differential increase, you could integrate wrt time and find the dynamic response.
Lost power=ext surface area* surface temp diff (with outside)*coeff of thermal convection.
I have neglected any temp gradient in the walls, which should be included in a lagged container. In this case take it as a conduction problem to outside with equivalent resistance to replace convection- ie inverse of convection coeff.